TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES. Remarks on topologically algebraic functors

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1 CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES REINHARD BÖRGER WALTER THOLEN Remarks on topologically algebraic functors Cahiers de topologie et géométrie différentielle catégoriques, tome 20, n o 2 (1979), p < 20_2_155_0> Andrée C. Ehresmann et les auteurs, 1979, tous droits réservés. L accès aux archives de la revue «Cahiers de topologie et géométrie différentielle catégoriques» implique l accord avec les conditions générales d utilisation ( Toute utilisation commerciale ou impression systématique est constitutive d une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

2 CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE Vol. XX - 2 (1979) REMARKS ON TOPOLOGICALLY ALGEBRAIC FUNCTORS by Reinhard BÖRGER and Walter THOLEN 1. INTRODUCTION. In order to describe functors appearing in universal topological algebra, Y.H. Hong [16] introduced the notion of a topologically algebraic functor. Although these functors are defined by a very simple factorization structure of cones, they allow to prove some important properties faithfulness, right-adjointness and lifting of existence of limits and colim its. such as Independently from Hong s notion the second author introduced the notion of an orthogonal m-functor (cf. [19, 20, 21] ) and proved general lifting theorems for adjoint functors extending related investigations for topological functors (cf. " yler l30]). In this paper a short proof is given that both concepts are equivalent: Each orthogonal m-functor is topologically algebraic, and for each topologically algebraic functo there is a largest 11 such that one has an orthogonal V-structure (Theorem 1). In [27], Trnkova introduced the notion of a (faithful) functor with «weak inductive generation» which was rephrased as «semi finally complete functor» by Wischnewsky [28] and as «semi-topological functor» by Hoffmann [14] and the second author [22]. These functors are treated more intensively in [23] and became interesting because of the following two main results proved here : Each semi-topological functor admits a so-called locally orthogonal factorization structure and, consequently, has a reflective topological completion, which turns to be its N4ac Neille completion (cf. Herrlich [9] ). It is known that topologically algebraic implies semi-topological (cf. [22]). Herrlich, Nakagawa, Strecker and Titcomb [11] and the first author [3] independently constructed countable counter-examples to the 155

3 reverse implication. In this paper we give a concrete counter-example, ie, we construct a semi-topological functor over Ens being not topologically algebraic (Theorem 3). Moreover, sufficient conditions for both concepts to be equivalent are given (Theorem 4). Semi-topological functors are those functors which have a reflective Mac N eille completion. By Herrlich and Strecker [10] topologically algebraic functors are characterised as those functors which admit a reflective universal completion. In order to give an effective construction of the universal completion of certain algebraic categories over 6;n6.. we prove a simple criterion for all P-initial cones to be mono-cones, if P is an arbitrary representable functor (Theorem 5 ). Besides Theorem 5 the results of this paper are taken from the preprints [4] and [24] which contain further investigations about topologically algebraic functors like external characterizations and lifting of monads and adjoint functors. But these results are omitted in this paper because they are immediate consequences or analogously gotten from corresponding results for semi-topological functors (cf. [25, 26] ). Furthermore, the reader is referred to the «Duality Theorems for topologically algebraic functors recently proved by Wischnewsky [29] and the theorems for lifting tensor products and inner hom-functors given by Greve [6] and Porst-"W ischnewsky [ 18]. 2. TOPOLOGICALLY ALGEBRAIC FUNCTORS AND ORTHOGONAL (&, M)-FUNCTORS. Let P : cr -+ X be a functor. A P-cone (over 9 ) is a triple (X, e, D) where X is an object of X, D: 9) -+ 8 is a functor and Z : D X -> P o D is a natural transformation whose domain is constant X. Often we write : Z : D X -> P o D category 1 instead of (X, Z, D). A P-cone over the one-morphism is called a P-morphism and written as a pair (x, A), where A is an object of à and x a morphism of 1( with codomain P A ; if X is the domain of x, we also write x: X - P A. The dual notions are P-cocone and P-comorphism. Note that objects of 1( can be regarded (co-)cones over 16. as P- 156

4 Epi(P) denotes the class of all P-epimorphisms e : X -> P A, i. e. for all u, v : A -> B with (Pu)e = (Pv)e one has u = v. Iso(P) denotes the class of all P-isomorphisms i : X - P A, i. e. i is an isomorphism of X. Note that Iso(P) C Epi(P ) holds if and only if P is faithful. Init(P) denotes the (meta-)class of all P-initial (1-cones v : D B -> D, i. e, for all 8-cones a : 0 A - D and all there is a un ique t : A -> B w ith P t = x and v (Dt) = a. DEFINITION 1 (Hong [16] ). P is called topologically algebraic iff any P -cone Z : D X -> P o D adm its a factorization w ith a P -epimorphism e : X - P A and a P -initia (t-cone 11 : A A -> D. Orthogonal (&, M )-functors are defined by assuming a diagonalization property instead of the condition that e has to be P-epimorphic. A P-morphism e : X - P A and an Q-cune v : A B - D are called orthogonal, written (e, A)1 (B, v, D), if for all P-morphisms x: X -> P B and all Q- cones a : D A -> D with one has a unique 1) t : A -> B with ( P t ) e = x and v (Dt) = a. For classes 6 of P-morphisms and 11 of d-cones define a Galois correspondence by 1) For functors considered here uniqueness follows necessarily: cf. Lemma

5 Obviously one has Now we assume 6 and 11 to be closed under composition with Q-isomorphisms, i. e. if i : A -> B is an isomorphism, let ((Pi) e, B )E & for all (e, A) E & and let (A, 03BC(Di))EM for all (B, 03BC) E M. DEFINITION 2 (Tholen [19, 20, 21]). P is called an orthogonal (6,YR)- functor if and if any P-cone Z : D X -> P o D adm its a factorization (*) with Because of Iso(P)1 = In it (P) one obviously gets m c Init (P). Moreover one easily proves the equations Therefore we also speak of an orthogonal 6-functor or an orthogonal m- functor, if the other parameter is not explicitly given. Tor-ologically algebraic functors and orthogonal (6, )R )-functor6 are faithful and right adjoint and lift the existence of all types of limits and colimits from X to Q (cf. [20,22] ). These properties hold more generally for semi-topological functors ( see 3 below). The connection between Definitions 1 and 2 is clarified by Theorem 1 below. For this we need the following Lemma 1 whose proof is based on a trick outlined more generally in [5] (cf. also [23], Corollary 6.4). L EMMA 1. I f P is an orthogonal (&, M)-functor, then & C Epi (P). PROOF. Let e : X -> P A be in and assume (P u)e = (P v) e, with u, v: A - B in 8. Let I = Mor Q and define a discrete P-cone It factorizes into (C, mi, Bi)I. Then, is not empty. "With any surjection a: I -> J define 158

6 Then there exists a m orphism t with mi t = hi, i E I, hence t E J, say t = a (io ). So we get THEOREM 1. The following conditions are e quivalent : (i) P is topologically algebraic. (ii) P is an orthogonal Init ( P )-functor. (iii) P is an orthogonal m-functor for some )R. (iv) P has a le ft adjoint and Q is an orthogonal m-category (i. e., IdQ is an orthogonal m-functor) for some subclass YR C Init (P). P ROO F. ( ii) => (iii) => (iv) is trivial and (iii) => (i) follows from L emma 1. So it remains to prove (i) => (ii). Let Z : D X -> P o D be a P -cone and consider all P-epimorphisms ei : X -> P Ai such that there is a P-initial cone tti: A Ai -> D with (iii is uniquely determined by ei ). The discrete P-cone ( ei : X - P Ai), has again an epimorphic and initial factorization Because of 1 # C and (e, A) E Epi (P) we can define ti = 03BCi(Dmi) independently from the choice of ie I. Hence we get a factorization (*) which turns out to be the desired one as can be shown in 3 steps. Step 1. 03BC : t1 A - D is P-initial. Proo f 1. For all B : A B - D and and any i f I one gets Therefore we have a t : B - A with P t = y. (03BC(Dt) =B and uniqueness of t follow from the faithfulness of P ). 159

7 Step 2. 1 f (Pm) e = e with E : X - P C P -epim orph ic and m : C - A P -initial, then m must be an isomorphism. Proof 2. m : C -> A in Init (P ) and 03BC : D A -> D in Init (P) imply 03BC(Dm) in Init (P), hence ê = ei with an i E I. Then the equations prove the assertion. Step 3, (e, A) E nit ( P)1. Proof 3. Let v : A B -> L be P -initial and let a : A A, E and be given. Then ( e : X -> P A, x : X -> P IJ) is a P-cone indexed by a two point set. Hence we get a P-epimorphism e : X -> cone (m : C -> A, n : C -> B ) with P C and a P-initial Ct. Because of Step 2 it suffices to show that m is P-initial as a single morphism ; then t : = n m -1 is a suitable diagonal with (Pt)e = x. Consider u : K -> A and y : P K -> P C with (Pm)y = Pu. Since v is P -initial, we get v : K - B with P v = (P n ) y, and since (m, n) is P-initial, we get w ; K - C with Pw=y. 160

8 R EMARKS. 10 Theorem 1 was proved independently and at the same time by Herrlich and Strecker [10]. Their proof is very different from the more «direct» proof given here. 2 Obviously Init (P) is the greatest class? such that a topologically algebraic functor P is an orthogonal m-functor. Correspondingly there is a smallest such that P is an orthogonal 6-functor, namely & = Init (P )1. Herrlich and Strecker [10] call P-morphisms of Init (P )1 semi-universal. 3. SEMI-TOPOLOGICAL FUNCTORS AND LOCALLY ORTHOGONAL (&, M)-FUNCTORS. Semi-topological functors arise from topologically algebraic functors very naturally. Namely, in the same way as Herrlich has done for (5; J m )-topological functors ( cf. [8], Lemma 6.1 ), one can prove the following «semi-final» property for topologically algebraic functors. L EMM A 2. Let P: Q -> X be a topologically algebraic functor. Then each P-cocone Z: Po D -> D X has a P-semi-final lifting>, a P-morp hism e : X - P A and an Q-cocone i. e. there exists such that for all y: X - P B and all f3: D -> A B with (A y)e = PoB there is a unique t : A, B with DEFINITION 3 (Trnkova [27], Wischnewsky [28], Hoffmann [14], Tholen [ 22] ). P is called semi-topological if any P-cocone admits a P-semi-final lifting. By the tduality Theorem* and the «Diagonal Lemma» proved in [ 23] it is possible to describe semi-topological functors also by a factorization property which is nearly as beautiful as the orthogonal one for topological algebraic functors (see Theorem 2 below). DEFINITION 4 (Tholen [ 23] ). and? be closed under composition with (î-isomorphisms. P is called a locally orthogonal (&, M)-functor if 161

9 Iso (P) C & and any P-cone Z : D Y -> Po D admits a factorization with p : Y - P B in & and it : D B -> D in? such th at, for a 11 e : X - P A in &, x: X -> Y in X and a: D A -> D with (Po a)(d e) = Z (Dx), one has a unique t : A - B with If fil functor. But note that 11 is not specified we briefly speak of a locally orthogonal &- is not determined by 6 or vice versa. Clearly, every orthogonal (&, M )-functor is a locally orthogonal (&, M )-functor. The precise connection between Definitions 2 and 4 is given by the following Lemma 3 which can be proved in the same way special Lemma 7.3 of [23]. as the little more LEMMA 3. The following conditions are equivalent: ( i ) P is an orthogonal 6-functor,. (ii) P is a locally orthogonal &functor and & is closed under composition, i. e. for all (p, A) E & and e : A - B in (t with (P e, B ) &, c one has ((P e)p, B)c 6. Very similarly to Theorem 1 one has the following characterization of semi-topological functors which is proved in [23]. There we denoted the class of all P-morphisms occuring in some P-semi-final lifting by: Quot (P). THEOREM 2. The following conditions are equivalent: ( i ) P is serrai-topological. (ii) P is a locally orthogonal Quot(P)-functor. 162

10 (i, e. (iii) P is a locally orthogonal &-functor for some &. (iv) P has a le ft adjoint and (t is a locall y orthogonal &-category IdQ is a locally orthogonal 6-functor) for some & C IVlor Q such that the counit of P belongs pointwise to 6. R EMARKS. 1 Quot(P) is the smallest such that a semi-topological functor P is a locally orthogonal 6-functor. 2 The P-morpbisms of Quot (P) are called P-quotients in [23] and semi-final in [l0]. For P faithful one has Quot (P) C Init (P)1 and the following «cone-free» characterization of each e : X - P A in Quot ( P ) : For any y : X - P B such that, for all there is a b : C - B with yx = P b, then there exists a unique 3 By M. B. Wischnewsky and the second author it was shown that every semi-topological functor P : d - X admits a factorization P = To E with T topological (= initially complete in the sense of Br3mmer [2] and Herrlich [9]) and E an embedding of a full reflective subcategory ( cf. [23] ). Both T and E are trivially topologically algebraic. Since compositions of semi-topological functors are again semi-topological (cf. [23]), in Cat the class of all semi-topological functors is the subcategory generated by all topologically algebraic functors. summarizes the results of Sections 2 and The following diagram 3 and shows which problems are to be investigated in the following. 163

11 4. A CONCRETE COUNTER-EXAMPLE. All examples of semi-topological functors given in [23] are already topologically algebraic. In the next section we shall prove the equivalence of both concepts under very mild additional conditions. Therefore, it is rather difficult to find semi-topological functors being not topologically algebraic. It is impossible to construct finite examples since Herrlich, Nakagawa, Strecker and Titcomb [11] have shown that a semi-topological functor P : Q -> X with 8 finite is already topologically algebraic. (Another proof for this can be given using the pushout characterization 6.3 of [23.) So they constructed a counter-example with categories having finitely many objects and countable hom-sets. Another example using countable preordered sets ( i. e. categories whose classes of objects are countable and whose hom-sets contain at most one morphism ) was independently constructed by the first author of the present paper (cf. [3]). Answering a question of J. Adamek we now prove : THEOREM 3. There exists a semi-topological functor P : Q -> &n6. being not topologically algebraic. P ROO F. Objects of S are triples (X, A, q5 ) where X is a set, A C X a subset and : 9 A - X a map defined on the power set of A such that A morphism f: (X, A, S) -> (Y, B, W) is given by a map f : X -> that f[a] C B and Y such P : Q &ns -> is obvious : P (X, A, 0) = X. I. P is semi-topological: It suffices to construct P-semi-final liftings of discrete P-cocones. Let (Yi, Bi, wi) be objects of ff, and let fi: Yi -> Z, i E I be maps. Define and consider the smallest equivalence relation - on V with the following properties: 164

12 , then N ow let s : Z -> v be the singleton map and p : projection map. Define q = p o s, A = q [ ] and We shall prove that (X, A, 0) and q yield a semi-final lifting. Step 1. K - q 1[q[K]nU for all KC U. Proo f l. Consider and apply property ( 2 ) of -. for all xc U because of Step 1. Trivially, Finally, for N C Bi we have Step 3. F or (Y, B, w) E Ob Q and g : Z -> Y w ith there exists a unique Proof 3. Define a map h : V - Y by h({z}) = g(z) for all ze Z and h(k) = w(g[k]) for all K C U ( h is well defined). In order to get f with f o p - h we have to show that the equivalence relation induced by h fulfills properties (1) and ( 2 ). This is easily done using the fact that all g fi o are morphisms of

13 go fi [Bi] C B for all if I implies f[a] C B. Furthermore, for M C A, one has M = p [q-1 [M] n U] and therefore for all K C U one has be any G-morphism with f o q = g. Then Together with II. P is not topologically algebraic : For all ordinals a > 2 define : Clearly, all (Y,, Ya, wa) E ObCl Assume there exists a P-initial d-cone such that I factorizes over Proof 1. Consider L et x E X and define Then all ha o g are morphisms of S, hence g itself, and therefore x = g(1) 166

14 is in A. Step 2. h, is surjective, Proof 2. If not, form. a singleton, since a > 2. Cbviously, ha[x] is not Therefore we get This is impossible. From the assertion of Step 2 we get card X > a for all ordinals a, so that our assumption must be wrong. R EM A RK S. 1 In Part II of the above proof we do not have to assume that ( {11} -> Ya)a > 2 factorizes over a P-epimorphism followed by a P-initial (1-cone. We could take any P-morphism and got the contradiction, too. But note that, if one generalizes Definition 1 in this way, one gets a properly greater class of functors. 2 The functor P, being semi-topological, has, in particular, a left adjoint which induces a monad on &n6-. Its Eilenberg -Moore algebras are just pointed sets. From Theorem 3 and Remark 3 of Section 3 we get: COROLLARY. 1 There exists a topological functor T: B -> &n6- and a full reflective subcategory Q of B such that T/Q is not topologically alge brai c. 20 Compositions of topologically algebraic functors need not be topologically algebraic, even i f their composition ends at &ns. REMARK. In [24], Corollary 31.4, the second author has studied the problem of cancellability of topologically algebraic functors : Let P = T o U be topologically algebraic such that Then U is semi-topological; moreover U is topologically algebraic, pro- 167

15 5. AN EQUIVALENCE THEOREM. Having the counter-example of Section 4 it is natural to look for conditions under which both, topologically algebraic and semi-topological, is the same. The conditions we give in the following are also sufficient to get a largest 6 such that one has a (locally) orthogonal &-functor, namely & = Epi(P). THEOREM 4. L et X be (co)complete and let (i be cowell-powered and well-powered with respect to extremal monomorphisms. Then the following conditions are equivalent: ( i ) P is semi-topological. (ii) P is topologically algebraic. (iii) P is a locally orthogonal E pi (P)-functor. (iv) P is an orthogonal Epi(P)-functor. (v) Q is (co)complete, P is faith ful and right-adjoint. P ROO F. One looks at the diagram and can restrict oneself to the first implication because the rest is wellknown. We have to distinguish two cases : S is cocomplete and Q is complete. The first case was considered in [23], Corollary 6.5: Q is a ( loc ally) orthogonal Epi(H)-category, hence, by Theorem 2 and L emma 3, P is an orthogonal Epi(P)-functor. Now let Q be complete. Let? be the class of P-initial extremal monomorphisms of (t. Because P is faithful, 11 will contain all regular monomorphisms, and? is closed under ( arbitrary ) pullbacks. So (t has (Epi(Q),m )-factorizations of morphisms fulfilling the diagonalization property. Now it remains to construct factorizations of cones from factorizations of morphisms. For this one uses the following lemma (an analogous result was proved in [20], Lemma 3.4), which completes the proof. L EMM A 4. Let Q have products and (&, M-factorizations of morphisms 168

16 fulfilling the diagonalization property with L et Q be &-cowell-powered, and let P be faithful and right-adjoint. Then P is an orthogonal &-functor with & containing all P-morphisms ( F being the le ft adjoint and E the counit). REMARKS. I In Theorem 4 the assumption of well-poweredness with respect to extremal monomorphisms is only needed in case X is complete. Generally, all the completeness and smallness assumptions of Theorem 4 are only needed to prove that d is an orthogonal Epi( (t) -category. This property also yields equivalence of conditions ( i ) - ( iv ) (cf. Herrlich, Nakagawa, Strecker, Titcomb [11] ). 2 One observes that we have proved in Theorem 4, (v) => ( iv ) that P is an orthogonal (Epi(P), M-functor with? C Mono(Q), where we denote by Mono(8) the class of all mono-cones it : A A, D, i. e., for one has u = v. From this fact alone it follows that 8 has coequalizers. (cf. [23], Proposition 5.3). This little observation proves once more Adamek s useful result on colimits of algebras (cf. [1]): COROLLARY. Let X be a cocomplete category, with (&, M)-factorizations o f morphisrrcs ful filling the diagonalization property with & C Fpi(X) and m C Mono(X) and let!( be &-cowell-powered. Then for every functor T : X -> X which the category o f T-algebras has coequalizers. PROOF. X is an orthogonal (6, 9 )-category with 9 C Mono (X). Because of T lil C l$ the same property holds for the category of T-algebras. 6. CATEGORIES IN WHICH EVERY INITIAL CONE IS A MONO-CONE. In [23], it is shown that every locally orthogonal factorization structure of a semi-topological functor P: Q -> X leads to a topological completion of P, i.e, there is a full reflective embedding E: (î and 169

17 a topological functor T : B -> X with T o E P. B = can be taken as a full subcategory of the comma-category Q, P> (whose objects are P-morphisms and whose morphisms are obvious commutative diagrams) with B = & In case = Quot (P) one gets the smallest topological completion, the Mac Neille completion of P ( cf. Herrlich [91, Porst [17] ). Herrlich and Strecker [10] show that for a topologically algebraic functor = Init ( P )1, one gets the so-called universal completion, which is just the reflection from the (meta-)category of faithful and amnestic functors over X with initiality preserving functors as morphisms into the full subcategory of topological functors over X. They prove that for the underlying set functor of a full epireflective subcategory of topological spaces, consisting of To-spaces and containing at least one space with more than one element, the Mac Neille completion and the universal completion are the same because of Quot (P) contains just all P-morphisms p: X -> P A with p surjective. We try to get an analogous result for monadic functors and, more generally, regular functors (cf. Herrlich [7] ) &n6.. over However, already simple examples like semi-groups and groups show that the situation is more complicated for «algebraic» functors. We are only able to give a sufficient condition which allows an effective computation of Init(P)1 and therefore of the universal completion of P. The key for this is the following more general Lemma 5 which seems to be interesting by itself. From [23] we repeat the notion of a P-semi-initial factorization that is the universal property of Definition 4 holds only with e the identity, i. e. for all P-comorphisms x : P A, X and d-cones one has a unique t : A -> B with 170

18 Clearly, if 03BC : DB -> D is P-initial, then (*) is P-semi-initial. A P-morphism e : X -> P A belongs to Quot(P) iff e allows no proper P-semiinitial factorization, i.e. if (Pm)q = e is a P-semi-initial factorization then m is an isomorphism (cf. [23]. L EMM A 5. Let P = (î (G, - ): (î -+ &ns be a representable functor such that G admits non-trivial epimorphic endomorphisms, i. e. there exists an epimorphism e : G -> G different from the identity: Let ( fi : A -> Bi), be a (discrete) Q-cone and let (P mi ) 0 cp = P fi be a semi-initial factorization of the P-cone (Pfi : P A - P Bi)I by an Q-cone (mi : C - Bi)I and a surjective map 0: P A, P C. Then ( P mi)i is a mono-cone, i. e. implies u = v. P ROOF. 1/e can assume P = (î ( G, -). Because of the semi-initiality 0 c an be written as 0 = P p with There are since 0 is surjective. Define a map y5 : P G - P A by Then hence (by definition of ta ) By semi-initiality, there exists a morphism One gets 171

19 and for all g # G. In case g = e this equation yields u = v. THEOREM 5. Let P = (I(G., -): Q -> &n6, be a representable functor such that G admits non-trivial epimorphic endomorphisms. Then every P-initial Q-cone is a mono-cone. PROOF. Application of Lemma 5 with 0 the identity map yields that the P-image of a P-initial 8-cone is a mono-cone, hence a mono-cone itself. (Note that P need not be faithful. ) Since mono-cones are P-initial for regular functors one gets : COROLLARY. Let P : Q -> &n6, be a regular functor and let G be the free o bje ct of Q with one generator. I f there exists an epimorphism e : G, G different from the identity, then Init ( P ) = Mono( ( (i ). Denote by Gen (P) the class of all P-generating morphisms g: X, P A, i.e. the induced morphism g : FX - A is a regular epimorphism ( F being the left adjoint of P ). COROLL ARY. Under the assumptions of the above Corollary, the objects o f a universal completion j3 of P are P-generating morphisms, and morphisms of j3 are given by commutative diagrams The topological functor B -> Ens forgets all of the diagram besides the upper row. PROOF. By the definition of a regular f unctor, P is an (Gen (P), Mono (Q))-functor, hence orthogonal EXAMPLES. 1 The underlying set functor of the following monadic cat- 172

20 egories fulfills the assumption of Theorem 5 :. ( abelian ) groups R-modules (char R # ( commutative ) rings ( commutative ) R-algebras 2 The following monadic categories contain initial cones which are not mono-cones : ( commutative ) semi-groups (see below), (commutative ) monoids ( see below ), pointed sets ( cf. Herrlich-Strecker [10] ), directed graphs (cf. Herrlich -Strecker [10] ). The example which can be -used in case of semi-groups is as trivial as intelligent. Consider the map and take the following table of composition : In case of monoids one has to adjoin a unit element. 3 The category Camp of compact T2-spaces is monadic &n&, over but does not fulfill the assumption of Theorem 5 (the one point space does not admit a non-trivial endomorphism ). Nevertheless, initial cones are mono-cones (and vice versa): Let (mi : B, Ai)I. Consider with be initial and assume Now, mi o w is continuous for all i E I, hence w, too, and x = y. The Mac Neille completion of Camp is properly smaller than its universal completion: Consider the dense (hence Init ( P )1- )map 173

21 from the (discrete) space N of natural numbers into its A lexandroff compactification and prove that j does not belong to Quot (P). It suffices to show th at j admits a proper semi-initial factorization. For this take the factorization of j over the Stone-Cech compactification of N. If there 0..-, is a map g: K -> N from a compact Hausdorff space K into N w ith j o g continuous, then g(k) must be finite and discrete, and (3N 0 g is continuous, too. Therefore the factorization is semi-initial. 4 Even for monadic categories &n6. over fulfilling the assumption of Theorem 5 the M.ac Neille completion can be properly smaller than the universal completion. For instance the categories of (commutative) groups and (commutative) rings contain a P-generating morphism which is not in Quot (P) : There is, namely, no map A - {1} such that the composition A - Z2 is an homomorphism since 0 does not belong to the image of this map. On the other hand, {1} -> Z2 is not universal, hence not the semi-final lifting of the empty P-coeone. 5 The category of Banach spaces over K ( K - R or C ) with norm decreasing maps as morphisms ( P being the unit-ball functor is not monadic over Ens but fulfills the assumption of Theorem 5 (x -> x is a non-trivial epimorphism of K ). The category monotone maps as morphisms of ordered sets with is not monadic over Ens and does not fulfill the assumption of Theorem 5, but initial cones are mono-cones, too (if (mi: B -Ai)I is initial and mi(x) = mi(y) for all ie I, consider the maps g, h: {0~I}-B with 174

22 REMARKS. 1 The sufficient condition of Lemma 5 that the representing object G admits a non-trivial epimorphic endomorphism can be obviously generalized: One only needs that all endomorphisms g of G different from the identity form an epi-cone, i, e. Then one gets many new examples for Theorem 5. For instance, let S be the Sierpinski space ( i. e. the only two-point space which is neither discrete nor indiscrete). Then the above condition is fulfilled although there is no non-trivial epimorphic endomorphism of S. Hence, with P = Top(S,-) P -initial cc.-ies are mono-cones. 2 He) 11ich and Strecker [10] characterized topologically algebraic functors as those functors which admit a reflective universal completion. Therefore, from Theorem 3 it follows that there is a functor P : A - Ens. admitting a reflective Mac Neille completion but not having a reflective universal completion. One can sharpen this result by proving that the universal completion of the functor constructed in Theorem 3 fails to exist ( in the same universe). Ilowever, to prove this one needs a stronger settheoretical assumption which, if K is a proper class and L is not empty, guarantees the existence of a surjection o : K - L. (Together with an axiom of foundation the following strong version of an axiom of choice is suff ic ient : There exists a function r with T ( X)E X for ali non-void sets X. ) For details the reader is referred to [4]. More generally the question of existence of universal completions is treated in a forthcoming paper by Adamek, Herrlich and Strecker ( cf. [31]). In case of the countable counterexamples given in [3] and [11] the universal completion, of course, does exist (but is not reflective), since all the categories in question are small (cf. [91 ). Therefore, the above set-based functor P is, in a sense, «less topologically algebraic» than the functors constructed in [3, 11]. 3 The functor P of Theorem 3 is ( strongly ) fibre-small and (hence) has a fibre-small Mac Neille completion. Therefore, besides completeness, cocompleteness, wellpoweredness and existence of a reflective Mac Neille completion, also fibre-smallness of the Mac Neille completion does not 175

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24 18. PORST, H.-E. & WISCHNEVSKY, M. B., Every topological category is convenient for Gelfand duality, Manuscripta Math. 25 ( 1978), THOLEN, W., Factorizations of cones along a functor, Quaestiones Math. 2 ( 1977), THOLEN, W., On Wyler s taut lift theorem, General Topology &Appl. 8 (1978), THOLEN, W., Zum Satz von Freyd und Kelly, Math. Ann. 232 ( 1978), THOLEN, W., M-functors, Nordwestdeutsches Kategorienseminar, Universität Bremen, THOLEN, W., Semi-topological functors I, J. Pure and Applied Algebra 15 ( 1979 ), THOLEN, W., Konkrete Funktoren, Preprint, Fernuni. Hagen, THOLEN, W. & WISCHNEWSKY, M. B., Semi-topological functors II : External characterizations, J. Pure and Appl. Algebra 15 ( 1979), THOLEN, W., WISCHNEWSKY, M. B. & WOLFF, H., Semi-topological functors III: Lifting of monads and adjoint functors. Fernuni. Hagen TRNKOVA, V., Automata and categories, Lecture Notes in Com. Science 32, Springer ( 1975), WISCHNEWSKY, M. B., A lifting theorem for right adjoints, Cahiers Topo. et Géo. Diff. 19(1978), WISCHNEWSKY, M. B., A generalized duality theorem for structure functors, Preprint, Universität Bremen, WYLER, O., On the categories of General Topology and topological Algebra, Arch. Math. 22, Basel ( 1971), ADAMEK, J., HERRLICH, H. & STRECKER G. E., Least and largest initial completions, Preprint, F achbere ich Mathematik Femuniversitat Postfach 940 D-5800 HAGEN, R. F. A. NOTE ADDED IN PROOF. "We thank the editors who drew our attention to two papers by C. Ehresmann dealing partly with related topics (Structures quasi-quotients, Math. Ann. 171 ( 1967), ; Prolongement universel d un foncteur par adjonction de limites, Rozprawy Mat. ( or Dissertationes Math.) G4 (1969)). 177